Dynamic equations for moments

Since the time when Osborne Reynolds [1] began his life of adventure in investigating the dynamics of turbulence, it has been postulated that the Navier-Stokes and continuity equations hold for the instantaneous values in turbulent flow: 2.1$$\frac{{\partial U_i }} {{\partial t}} + U_k \frac{{\partial U_i }} {{\partial x_k }} = - \frac{1} {\varrho }\frac{{\partial P}} {{\partial x_i }} + v\frac{{\partial ^2 U_i }} {{\partial x_k \partial x_k }},i,k = 1,2,3$$2.2$$\frac{{\partial U_i }} {{\partial x_i }} = 0.$$ We may ignore speculations presented in the literature about the validity of this fundamental postulate; there exists firm evidence accumulated from experimental observations and numerical simulations which justify the validity of the Navier-Stokes equations for application to instantaneous motions in turbulent flow.